A : To a metal wire of diamete d and length L when the applied voltage is doubled, drift velocity gets doubled.
R : For a constant votage when the length is doubled, drift velocity will be halved but drift velocity is independent of diameter .

Options

A. If both Assertion & Reason are true and the reason is the correct explanation of the assertion , then mark
B. If both Assertion & Reason are true but the reason is not the correct explanation of the assertion then mark.
C. If Assertion is ture statement but Reason is false , then mark
D. If both Assertion and Reason are false statements then mark.

Answer

To solve the question, we need to analyze the assertion and reason provided regarding the drift velocity in a metal wire when the applied voltage and length are changed.

### Step-by-Step Solution:

1. **Understanding Drift Velocity**:
The drift velocity \( V_d \) of charge carriers in a conductor is given by the formula:
\[ I = n e A V_d \]
where:
- \( I \) is the current,
- \( n \) is the number density of charge carriers,
- \( e \) is the charge of an electron,
- \( A \) is the cross-sectional area of the wire,
- \( V_d \) is the drift velocity.

2. **Relating Current to Voltage**:
The current can also be expressed using Ohm's law:
\[ I = \frac{V}{R} \]
where \( V \) is the voltage and \( R \) is the resistance.

3. **Substituting Resistance**:
The resistance \( R \) of a wire can be expressed as:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area.

4. **Combining Equations**:
By substituting \( R \) into the current equation, we get:
\[ I = \frac{V}{\frac{\rho L}{A}} = \frac{VA}{\rho L} \]
Now, substituting this expression for \( I \) into the drift velocity equation:
\[ n e A V_d = \frac{VA}{\rho L} \]
Simplifying this gives:
\[ V_d = \frac{V}{n e \rho L} \]

5. **Analyzing the Assertion**:
- When the applied voltage \( V \) is doubled, the drift velocity \( V_d \) becomes:
\[ V_d' = \frac{2V}{n e \rho L} = 2V_d \]
Thus, the assertion is correct: doubling the voltage doubles the drift velocity.

6. **Analyzing the Reason**:
- For a constant voltage, if the length \( L \) is doubled, the new drift velocity \( V_d' \) becomes:
\[ V_d' = \frac{V}{n e \rho (2L)} = \frac{1}{2} \cdot \frac{V}{n e \rho L} = \frac{1}{2} V_d \]
Therefore, the drift velocity is halved when the length is doubled.
- Additionally, the drift velocity is independent of the diameter since the area \( A \) (which is proportional to \( d^2 \)) cancels out in the equations.

7. **Conclusion**:
Both the assertion and reason are correct. Therefore, the answer is option 2.